ECONTECHMOD. AN INTERNATIONAL QUARTERLY JOURNAL – 2018, Vol. 07, No. 2, 59-69

Optimal laws of gear shift in automotive transmissions

P. Hashchuk1, R.Pelo2

1Lviv State University of Life Safety, e-mail:

2National University “Lviv Polytechnics”, e-mail: peliorom70@gmail.com

Received July 07.2017: accepted Marc 10.2018

Abstract. In the article the influence of the moment of

beginning of shifting gears on the efficiency of acceleration

of the vehicle has been evaluated - on the fuel consumption

at a given level of dynamism of the car. Different programs

of switching gear at different values of the length of the

friction trailing have been studied. Arguments are made in

favor of the fact that the gearshift processes should not be

too fast and that the work of friction trailing is not decisive

in view of the need to increase the energy efficiency of the

car. It is emphasized that between the laws of gear shifting,

which are optimal considering the fuel economy, which,

having in mind the dynamism, there are no fundamental

differences. On the basis of the information obtained, it is

concluded that the real variety of transmissions is

unreasonably excessive and even harmful, since optimality

is usually monotonous.

Key word: automobile, transmission, gear shift,

optimal laws, transmission excellence, control algorithms,

optimality criterions, time acceleration, fuel consumption.

INTRODUCTION

The diversity of existing designs of automotive

transmissions and their laws and algorithms are extremely

broad [1-12]. But this, of course, is not a sign of the

remarkable achievements of engineering thought and

technical science, since any variety normally reduces itself

to the search for anything perfect or optimal.

It is believed that the automatic gearbox was invented

due to the intellectual efforts of the agents of the American

concern General Motors. The first car with an automatic

transmission - Oldsmobile Custom & Cruiser - descended

from the conveyor in 1939. Since then, apparently, there

were grounds for measuring the principles, laws,

algorithms of expedient or optimal control of the structure

and parameters of automotive transmission.

Typical automatic transmission with a hydrodynamic

(predominantly complex) transformer (or hydraulic

coupler) usually "proposes" for the choice of the driver

himself the following modes of operation: P ("Park") -

parking (wheeled running gears are locked, but the lock is

done internally in gearboxes and not connected with the

parking brake); N ("Neutral") - an idle run; R ("Reverse")

– back run; D ("Drive") - movement (forward, of course;

all forward programs can be used, all except for upgrades);

L ("Low") - lower gear, slow down (in case of difficult road

conditions) . The choice among these modes is exercised

by the driver with the lever-selector of the mode ranges.

But the selection of transmission and the actual process of

switching gears from one to another carries out a system-

automatic machine (in our time - with the participation of

the on-board computer). The system also imposes

restrictions on passive safety: for example, it will not be

possible to start the engine when the selector lever is not in

the "P" and "N" positions. It also makes it impossible for

the self-propelled movement of a car while parked on a

non-horizontally or locally unequal platform (it is possible

to remove the key from the ignition lock only if the lever-

selector is switched to the "P" position). At the same time,

the "P" mode does not replace the handbrake.

But so many special modes of auto-transmissions,

which the driver has to pick up independently, is rather a

sign of its excessive complexity. Intectualization of

electronic control systems has created the basis for the

provision of automatic transmissions of special properties.

In particular, there appeared so-called adaptive automatic

gearboxes. The adaptability is that the on-board computer

monitors the driver's way of driving the car and adjusts to

it. The algorithm of computer operation at times even

assumes control over the degree of wear of frictions.

But what is so useful in adaptability? If the machine is

perfect, why should it be adjusted to the driver? If the

driver is smarter than an automatic machine, then what is

this machine for?

RECENT RESEARCHES AND PUBLICATIONS

ANALYSIS

There are automatic gearshift control systems -

AutoStick (Steptronic, Tiptronic) - that give the driver the

option of the command itself to choose gears, but they are

entirely responsible for the process of shifing. In this case,

the lever-selector has additional mode settings. But the

independence of the driver is sometimes illusory, because

the Autostick mode is not less automatic: the computer

system still "will not allow" to directly affect the unit; the

driver only sends his wishes to the computer, and it

analyzes their adequacy and makes decisions about

switching (to move, for example, from a third transmission

or to turn on the transmission, at which modes of the engine

will leave the set of permissible ones will not succeed). In

everything else, the transmission is like an usual one,

mechanic. Of course, the driver at any moment can move

the selector lever to the "D" position, refusing to act as if

directly controlling the transmission.

60 P. HASHCHUK, R.PELO

But what for the developer to invest in the intelligence

and means in automation, and then also in the means of

simulating non-automatic ("manual")? From what such a

miracle (rather than ignorance), there is such a subjective,

unreasonable demand in the market?

It is also introduced, in paralle, several transmission

control algorithms for the drivers’ choice - energy-saving,

sports, winter. The energy-saving mode seems to be

designed to ensure the smooth movement of the car with

minimal fuel consumption (which, of course, interpretation

and energy-saving meters are usually questionable). The

sports algorithm is configured to realize the maximum

engine power and, accordingly, the maximum acceleration

of the car. There is also a kickdown mode, when in the case

of a sharp press to the edge of the accelerator pedal, the

system switches the transmission to a lower gear - one or

one through one. Reverse switching again to a higher gear

can occur if the engine reaches the maximum frequency of

operating cycles. The winter algorithm foresees the

possibility of a smooth start grip of a car on a slippery

surface (usually a start should take place on the second or

third gear).

But is there any certainty that these algorithms are

really needed and that the driver is capable of handling

them rationally? Is this variation not controlled by the

automatic system? And can not the problem of choosing an

algorithm be solved on the basis of a compromise, at least

partial? Or maybe the motivation of polyalgorythmicity is

fictitious? It is on this that we will have to focus further.

The automatic transmission is less energy-efficient (its

performance index is generally lower) compared to the

mechanical transmission, but due to the optimal

combination with the engine and the implementation of

optimal blocking laws for the hydrodynamic transformer,

as well as the laws of gear shifting in its mechanical part,

it is potentially capable of still sometimes provide higher

fuel economy and dynamism of the car. And yet: the

property of an automatic transmission to absorb shock

loads contributes to an increase in engine resource and

undercarriage.

But this does not mean that the classic automatic

transmission does not have an equivalent alternative.

Rather, on the contrary, it outlines the possible directions

for the improvement of mechanical energy transformers.

So, to improve the traditional mechanical transmission

becomes possible, introducing the principles of optimal

combination of modes, optimal control laws, means of

increasing the elasticity - all that the automatic

transmission is attractive. It is interesting that in the

transmission of the Mercedes AMG Speedshift MCT 7, the

hydraulic transformer was replaced by a "fluid" friction

clutch - this made it possible to significantly increase the

efficiency of the transmission and combine it with high-

speed engines.

Variator (belt, chain, torque ...) is a transmission that

has an infinite number of gears (levels). It can implement

any transfer ratio from its operating range and change it

smoothly so that the rotating torque and speed of the output

shaft will change accordingly, even if the engine shaft

rotation frequency is constant, corresponding, for example,

to nominal power . The variator is not capable of properly

displaying its positive properties without a perfect control

system. It was with the development of microelectronics

that the variator attracted persistent attention. The on-board

computer is instructed to coordinate the work of the engine

and the transmission, achieving at each moment the

optimal transfer ratio. It is also instructed to inculcate even

absurd modes. For example, here and there they succeed in

simulating the step-by-step gearbox in order to please the

capricious driver who is poorly aware of what is good and

what is not, and he insistently wants to shift the

transmission on his own; in the memory of the control unit,

there several transfer relations are recorded that the driver

chooses (the Tiptronic principle is a marketing trick that

actually only worsens the energy-efficient and dynamic

transmission properties). There is also an analogue of the

kickdown in the variator: the stroke on the accelerator

pedal to the edge (if desired, to accelerate the car) generates

a sharp increase in the torque at the output of the variator.

The traditional mechanical transmission has in its turn

a friction clutch, a mechanical gearbox, in which the driver

at his discretion by means of the lever chooses the

transmission and makes the switching from the

transmission to the transmission through the clutch and the

mechanism of switching with the synchronizers (once a

long time without them) . The skillful driver changes the

transmission in a synchronized box for 0.5 ... 0.6 seconds.

The first mechanical boxes "contained" two transmissions

- one for starting the car, the second for movement.

Nowadays - dozens of gears, as if a means to approach the

variator. But the primitive increase in the number of gears

in a mechanical transmission is a deadlocked path, because

the transmission loses to a certain extent and, in a certain

sense, controllability.

Of course, the first thing that naturally comes to mind

- the traditional mechanical transmission to provide a

machine that would itself select and switch gears. Such an

operation is called automation or robotization. Accordingly

- automated (automatic) or robotizated (robotic) – so will

have to call such improved transmission. It is also referred

to as the Manual Transmission Automatically Shifted or

MTA (Automated Manual Transmission) abbreviation.

The path of robotization can be different. It is worth

mentioning that among the first not always successful steps

in the way of automation was the automation of only

coupling. As an example, Toyota's FreeTronic (TFT) is

very unreliable, it's purely mechanical, but with automatic

coupling. In the mechanical transmission of the Mercedes-

Benz A-Class car, for example, an automatic (electro-

hydraulic) coupling device was also installed. The

gearshift is placed entirely on the driver, and in order to

help him, the electro-gliding machine must track the

current position of the gear lever control (selector) and

carry out the necessary manipulations with the clutch. The

electronic control system must take into account the signals

of the engine and ABS sensors and prevent the jerks when

switching gears and prevent the engine from dampening. If

we limit ourselves to automating only the coupler, then we

will deal with a semi-automatic transmission. In the case of

partial or complete automation, the coupling control pedal

disappears, as in the case of an automatic gearbox - the

coupling is controlled by the machine.

As an example of deeper robotization we can present

the adaptive robotized gearbox 2-Tronic, developed by the

OPTIMAL LAWS OF GEAR SHIFT IN AUTOMOTIVE… 61 French group PSA Peugeot Citroen in conjunction with

Siemens and Bosch for the (Peugeot 207) car: the

transmission has ripened on the basis of a five-stage purely

mechanical, which was made already two decades ago ;

two electromechanical drives were attached to it, one of

which makes gear shifting, and the second switches on and

off the coupling, as well as an on-board computer that

manages these processes based on information on the

values of different modes of operation, first of all - about

the position of accelerator and the speed of movement of a

car. Simplified, with a single clutch, robot (we mean, for

example, the transmissions of cars Toyota, Opel, Alfa

Romeo, Peugeot, Suzuki) makes a gear shift in 1 ... 2

seconds. It remains possible for manual gearshifting. The

2-Tronic gearbox, for example, provides the ability to use

three modes: the first one - fully automated; the second is

the so-called semi-mechanical, which can be used without

leaving the fully automatic mode, in the case of urgent need

to switch to the lower gear (in the process of overtaking,

let's say, when the situation has run out and the car has

returned to normal mode of movement, the gearbox after a

while unwittingly restores its automatism); the third - quite

manual (though, if the driver will prove the speed of the

engine shaft to the maximum allowable value, without

switching the transmission to a higher gear, then it will still

make it for him).

Recognized for high-quality robotics, for example, the

six-speed manual transmission of the BMW M-series,

which is called SMG-Sequential M Gear-box (sequential

transmission from the transmission to the transmission,

from Latin sequentia – following up). The switching of the

coupler and the gearshift is placed on an electronically

controlled hydraulic system. The speed of switching of the

gears is very high, during acceleration of the car it takes

0.08 seconds.

The computer-aided intellectualization of the

automaton-robot reveals the way to a higher level of

perfection of the robotizated transmission in comparison

with the parametric same traditional mechanical

transmission. The high-quality electronic control of

actuators allows even the refusal of so-called

synchronizers. Due to this AMT becomes in a linear

dimension (in length) more compact and also capable of

sending big rotary moments, it will need less oil, and even

the weight will decrease. Robotized gearboxes may have

either an electric or hydraulic coupling and a gear

changeover mechanism. In the case of an electric drive,

servodevices are the executive bodies (electric motors),

and in the case of a hydraulic drive - hydraulic cylinders. It

so happened that in the case of a hydraulic drive (which

happens more often), the gearbox is sequential (it is a

matter of sequential gear shifting in manual mode).

Generally speaking, potentially there are ways to

technically improve a robotic transmission to such an

extent that it can compete both in the dynamic transmission

of energy and in energy-saving with any conventional

mechanical and with any automatic (either traditional or

variational). In this case, the usual problem of overheating

of the clutch will remain in the past.

A fundamentally deeper refinement overcomes a

robotic transmission, if it is to be introduced into one more

coupler. The most famous among them is the transmission

with the so-called DSG transmission box (Direct Shift

Gearbox, sometimes - Dual Clutch Transmission), which

was used on cars Volkswagen and Audi. It came to a wide

world from motorsport (this is a technology from Formula

1). Transmission DS (G) was massively used on

Volkswagen's Golf R32 cars in 2002/03. It is tried to be

applied even on tractor technology [12].

So, one clutch should serve uneven gears, and the

other - even. If, for example, the car's motion occurs on the

third transmission with the closed first coupler (second

off), then if necessary, switches on the pre-configured by

computer machine the fourth gear (with an increase in

speed) or a pre-configured - the second (in case of speed

reduction), the first coupling turns off (and goes into the

stand-by mode), and the second synchronously turns on.

The gearshift takes fractions of a second. For example,

gearboxes with two DSG, S-Tronic couplings are switched

from transmission to transmission for 0.2 ... 0.4 s, SMG

and DCT M Drivelogic boxes of sports cars BMW - for 0.1

s.

In this case, it can be said that the transmission

contains a pre-selective gearbox: after switching on any

transmission, you can pre-select the next one and activate

it at the right moment without interrupting the energy flow.

Transmissions can be switched without loss of power.

Potentially, the DSG transmission, compared with all of

the rest, can provide the car with the highest both

dynamism and energy efficiency. Of course, in the DS (G)

-ransmission from an odd one, let us say, to the odd one

can be switched only through an intermediate even one.

Instead, the automatic transmission with planetary gears

rows friction clutches can provide a random switching

sequence, for example, the Mercedes 7G-Tronic can jump

in one step into four transmissions "down". But it is

unlikely that this can be considered an advantage of an

automatic transmission.

In the DCT M Drivelogic dual-clutch transmission

from BMW, the Drivelogic features a control system that

allows for the use of eleven gearshift programs. Six

programs are implemented in manual switching mode, and

five are programmed for automated gearshifting. So there

is an opportunity to adapt the conditions of the change of

gear according to the style of driving. But this kind of

programmatic diversity and adaptability are evidence of

imperfections (let's mention the adaptability and

combination of energy-saving, sports, winter control

algorithms).

The more powerful the engine with which the DSG

transmission works, the greater the energy losses in the

vicinity of the so-called kiss point, when both couplings are

simultaneously intensively trailing. This is why the new

seven-speed gearbox DSG with two dry clutches,

developed jointly by Volkswagen and the Luk companies,

should not send a torque greater than 250 Nm. Instead, a

longer six-speed DSG gearbox with two fluid clutches can

work with a larger engine (clutches in a fluid crankcase,

frictions). But the energy loss during the switchover is

inevitable, they accompany switching in any other

transmission.

It is believed that in order to completely eliminate a

person from the control circuit AMT should

algorithmically cooperate with some hundreds of sensors

62 P. HASHCHUK, R.PELO that would have to supply an automatic system with an

adequate amount of information of the proper quality. Such

a large number of constituents, of course, negatively affects

the potential reliability of the control system. Inadequate

work of one sensor causes distortion of control algorithms:

energy saving, dynamic and any other algorithm cease to

be such, but rather transform into non-working,

emergency... Although there are quite effective computer

tools for a detailed study of the energy performance of a

car and an engine [13, 14], it is also possible to assume that

the optimality of the laws of switching gear is substantially

simpler to rely on such a large amount of instrumental and

computer information [15-18].

OBJECTIVES

The purpose of the research: to find out how much the

existing diversity of the laws of mechanical gearbox

transmissions control is motivated; determine how deeply

different criteria of optimality of the laws of switching the

steps (levels) contradict each other; to assess whether there

is a sense in parallel to foresee several transmission control

algorithms.

PRESENTATION OF THE MAIN MATERIAL

Task and scientific hypothesis. The attitude to

different transmissions can be substantiated if it is possible

to answer impartially such questions.

1. What should be the ideal gear shifting time? Is

transiency really relevant to the gear shift (remember, in

the Dynamic mode, some transmissions change the

transmission for 0.08 s and this is considered to be a huge

success)?

2. What parameters determine the expediency of using

one or another transmission at one time or another? Should

the number of sensors, that supply the information to the

micro-roprocessor control system of the transmission, be

almost one hundred?

3. What distinguishes the special energy-saving and

dynamic (sports) transmission laws? Are they essential to

each other and must necessarily be programmed in a

perfect step-up transmission?

4. Is the kickdown on the driver's share of the

accelerator pedal on the floor? Is this a truly special

transmission control mode that would have to oblige the

gearbox to accelerate the acceleration of the vehicle by

switching to a lower gear or through one down.

5. Has a mechanical shift with manual switching the

objective chances to hold on the market as an attractive

product? At the very least, it is appropriate to say about

these chances if optimal transmission control laws were

easy to reproduce (at least with an acceptable accuracy)

without automation.

The scientific hypothesis can be put forward in the

form of a statement that for all of the listed questions there

are reasons to give a warning or a completely negative

answer.

About Optimal Laws Of Gear Shift. For the

implementation of the procedure for the synthesis of

optimal gearshift laws, the known information is, as is

known, the parameters of the gearbox and fuel

characteristics of the engine ),( eett МQQ [19-22],

where tQ - the speed of fuel consumption,

eM - the

torque, e - the speed of rotation of the engine shaft;

eee NМ

- engine power performance. The special

features of the process of synthesis of optimal transmission

laws without interruption of the power flow should be

monitored on an example of a simple mechanical gearbox

with friction controls in case of realization by a given car,

let us notice again - a simple program of motion of a

vehicle.

Let us reproduce in a certain interval ],[ BA tt of time

t a fragment AB of the program )(tVv of motion of a

car with a constant acceleration (fig. 1: v - speed). The

assumption of the linearity of the program )(tVv is quite

acceptable [19, 20], since the process of gear shifting is

short and hence the vehicle's motion is considered during a

very small time interval. This fragment of the program of

motion of the vehicle can be fully or partly implemented,

forcing the engine to work on the sets of modes represented

by some curves of the dependence )( eeke NN or

)()1( ekee NN of power eN of the engine on the speed

e of rotation of its shaft on certain k-th and (k+1)-th gears

(Fig. 2). The lines )( eee NN , )( eee NN ,

const

ee , const

ee reflect in the

coordinate system eеON

of the so-called external modes

of operation of the engine and encircle a set of possible

modes of operation of the engine. As a matter of fact, let

this set of external include the modes and operation At

kR

and Bt

kR 1 of the engine, corresponding to the A beginning

and B the end of the given program of motion of the vehicle

(corresponding to the momentum At and Bt , (see Fig. 1).

Fig. 1. Fragments of the program of movement of the

car

In this case, the chosen vehicle program )(tVv can

be implemented, using necessarily two transmissions (two

levels) of the transmission; the gearshift from k-th to (k+1)-

th should occur when the engine reaches on k-th gear some

mode 0t

kR on the line )( eekN , see Fig. 2.

Of course, the driver chooses program, guided by the

conditions and circumstances of the traffic. And, of course,

OPTIMAL LAWS OF GEAR SHIFT IN AUTOMOTIVE… 63 the machine must not adjust the driver's choice. But the

deviation )(tVV or

)(tVV N from the given program

)(tV of motion (see Figure 1) is possible, however, due to

different technical constraints. But it is possible to put

forward a logical requirement that for a moment

Stt (

SS tt

or N

SS tt

, see fig. 1) after switching gears the

consequences of the deviation of the traffic program could

not be found. And this means that conditions must be

fulfilled at this moment

SS t

t

t

t

dttVdttVS

00

)()(

and

)()()(

SSS tVtVtv

- Control: BA t

kk

t

k

t

k

t

k RdeRabcRRR 11100

g- Control: BA t

k

g

k

t

k

t

k

t

k RdeRabcRRR 11100

а б

N- Control: BA t

k

N

k

t

k

t

k

t

k RdeRabcRRR 11100

M- Control: BA t

k

M

k

t

k

t

k

t

k RbcRaRRR 11100

в г

Fig. 2. Combination of operating modes of the engine and a step-by-step transmission of the car

Parameters characterizing the process of gear

shifting in the step-by-step transmission of the car. Let

the current mode R of the car engine (see Fig. 2), moving

to the right along the curve )( eeke NN (a certain k-th

gear is turned on), will turn into a mode 0t

kR , and at this

moment should begin shifting gears to (k+1)-th. The upper

symbol in the designation 0t

kR indicates precisely at the

moment of the program of movement of the car, which

coincides with the beginning of the gear shifting process;

therefore, the symbol 0t in the designation 0t

kR indicates

that the moment of the beginning of the re-transmission of

the transmissions from k-th to (k+1)-th coincides with the

moment 0t , indicated on the program of motion )(tVv ,

see. Fig. 1. Actually, at the moment 0t frictions k and

1k switching devices start to operate (Fig. 3; eI

moment

of inertia of the transmission masses associated with the

primary shaft of the gearbox; aI - the instantaneous

moment of inertia of the transmission mass between the

gearbox and the main transmission); and so that after a

certain fixed time of full (k+1)-th gear transfer, the

current mode of the engine coincides with the mode

0

1

t

kR

, which belongs to )()1( ekee NN curve. The lower

index in the designation

0

1

t

kR indicates the number of gear

after switching transmission, and the upper symbol -

0tt - the moment of switching completion; -

duration of the gearshift process (in the sense of purely

switching-trailing of frictions).

The process of changing the mode of operation of the

engine due to switching over the transmission time with

frictions k and 1k depends on the method of engine

control. As soon as the engine's operating state reaches the

64 P. HASHCHUK, R.PELO

switching position

0

1

t

kR , the implementation of the

program of motion of the car will be complete so that the

current mode R will move along the curve )()1( ekeN ,

until a completely definite position Bt

kR 1 which

corresponds to the moment of completion of the controlled

part of the program of movement of the car.

Fig. 3. The scheme of combination in the transmission

of a car of two levels, one of which – the direct

transmission

Of course, in the moment 0t of the switching gear

start, another arbitrary moment it of the reproduction

process of the given program of the car's motion can be

taken arbitrarily, and instead of the quantity 0t

corresponding to the moment of completion of the work of

the frictions, it is possible to give preference to some other

acceptable value jit . Therefore, it seems quite natural

the task of finding the optimal values of the parameters it

and jit , as well as the optimal (or at least rational)

method of engine and friction control.

Work of frictions. Frictions k and 1k each time

together must provide a transition it

kR - jit

kR

1 of the mode

R of operation of the engine from the curve )( eekN to the

curve )()1( ekeN without any violation, as noted, the

given by the driver of the program of motion of the car.

But even this particular task can be accomplished,

embodying a variety of programs for trailing friction

elements. In particular, it may be required that the gearshift

be carried out either in a predetermined time interval,

subject to a particular requirement, or at the shortest time

(dynamic switching with limited friction resources), or

with the least energy dissipation (energy saving switching),

or else.

In accordance with the scheme given in Fig. 3,

)1(1 kkkk

eee MMM

dt

dIM

, (1)

ak

k

k MMu

M

1

, (2)

where kM

and

1kM - rotating moments created by

frictions k and 1k ; 0ekM , 01 ekM ; ku -

transfer ratio (k-th gear). Since the program of the car's

motion is given, then it is also known the quantity at every

moment of time (see (2))

)()()(

1 tMtMu

tMak

k

k

. (3)

Consider the algorithm for synthesizing the optimal

control laws of the step-by-step mechanical transmission

of a vehicle, not taking into account the energy loss on

friction in toothed catching and bearing shafts. In addition,

let's put into (1):

battMMM kkkk )()1(1

, (4)

where the coefficients a and b are determined by the

values of the predefined parameters it , jit and the

coordinates of the points

i

i

i t

ek

t

eke

t

ekdt

dIM

, ,

ji

ji

ji t

ke

t

ke

e

t

kedt

dIM

)1(

)1(

)1( ,

respectively ( it

ekM , it

ek , - the coordinates of the point

it

kR ; jit

keM

)1( , ji tt

ke

)1( - the coordinates of the point jit

kR

1 ).

Expression (4) peculiar reflects the program of compatible

work of two frictions in the process of switching gear from

k-th to (k+1)-th.

Thus, the relations (3) and (4) together clearly define

the programs )(tMM kk

and )(11 tMM kk

of the

work of frictionsk and

1k ; (through parameters a , b

, ku , time t, and function )(tMM aa ):

k

k

ak u

u

battMM

1

)(

, k

kak

u

utMbatM

1

)(1

.

The function )(tMM aa is known, since it is

uniquely determined through the program of the car

movement (except, of course, that time interval when there

are technical restrictions):

t

tvmtFvkG

urtM a

d

)(d)()( a

2

пa

тр

0к

,

where 0u - transfer ratio of the main transmission; тр

-

coefficient of efficiency of the transmission; кr - wheel

radius (we believe that the radius of rolling and the

dynamic radius are one and the same concept); aG - weight

of the car; - the total coefficient of resistance of the road;

пk - coefficient of air resistance (aerodynamic resistance);

F - area of wind resistance (sail area); am - mass of the car;

- coefficient of account of the inertia of the rotating

masses of the car, which is determined by the formula

2

кa

к2

кa

тр

2

0 11

rmI

rm

uIa

,

where кI - the total moment of inertia of the wheels.

It should be noted that compliance with a given

program of motion throughout the switching time is

possible only under exceptional conditions. The speed of

the sliding (trailing) of frictions and (fig. 3) is determined

by the formulas respectively

OPTIMAL LAWS OF GEAR SHIFT IN AUTOMOTIVE… 65

aeeeskz

z

z

z

z

z

z

z

1

2

3

44

1

2

3

41

aeeks 4)1( ,

where 1 ,

4 - the speed of rotation of gears with the

number of teeth; 1z ,

4z ; 2z and

3z - the number of teeth

on the gears of the intermediate shaft (which speed is -

23 ).

Let us assume that 0aM (that is, the deceleration of

the car by the engine does not matter). Therefore, the

conditions must also be met

0e , 0a , 0)1( aeks (5)

(in case 0)1( aeks

the condition

0/ kaesk u ( 1ku ), would be true, which

would mean braking by the engine). Quantity

kaesk u/ ( 1ku )

in general, it can be both positive and negative. Therefore,

the transmission of torque in both frictions is possible when

0

k

aekskk

uMM

,

0)(1)1(1 aekksk MM ,

preventing the ability to receive energy from the outside

(frictions can only dissipate energy).

Ways to control the engine. In the process of trailing

frictions, the engine operation mode R can move in the

coordinate system eеON along the lines At

kR 0t

kR a b c

0

1

t

kR d e

1kR Bt

kR 1 (Fig. 2, a); At

kR 0t

kR a - b c

0

1

t

kR d e

g

kR 1Bt

kR 1 - (Fig. 2, b); or lines At

kR 0t

kR a - b c

0

1

t

kR d e

N

kR 1Bt

kR 1 - (Fig. 2, c). In this case, in first two modes of

control, the engine is "obliged" to implement at the the

beginning of the work of frictions (at any moment 0t ), the

forced modes (areas ab), then unforsed (areas c

0

1

t

kR ), and

from some moment 0t again forced (areas de). For the

third mode, forced modes are implemented throughout the

entire control. However, the implementation of the final

stage of control "requires" from the engine to provide: in

the case of the first mode of control - the constant angular

velocity (vertical section c

0

1

t

kR ); in the case of the se

cond - the section c

0

1

t

kR belongs to the line of minimum

specific fuel consumption )(0

eee NN . In these three

control options, the total duration of friction work will be

considered the same - . Therefore, at the point of the

mode

0

1

t

kR the trailing of frictions stops and switching

gears can be considered conditionally not completed. The

first way to control the engine, let us call it -control (by

the sign e = const), the second - g -control (on the basis

of compliance 0

eN modes with the minimum value of

specific fuel consumption )/( eete MQg ), and the third

- the N -control (or dynamic control - on the basis of the

full independence of the external modes of the engine

)( eee NN and )( eee NN ). In fig. 2, d shows

another version of the engine control - At

kR 0t

kR a

0

1

t

kR - b

cM

kR 1Bt

kR 1 . In this case, the engine immediately, from the

moment of the start of the friction "is obliged" to

implement a constant torque. This way of controlling the

engine 0t and for these parameters does not involve

external modes, and therefore does not belong to the forced

ones. We will call it (by the sign eM = const) M-control.

Note, that the duration 1 (see Fig. 5) of the passage

of the sections ab (see Figures 2, a and b) and the section

0

1

t

kcR (see Figure 2, c) are small compared with the

duration of the whole process of frictions’ work. The

interval of time 1 corresponds to the areas of falling of

angular speed of the engine 1 (see Fig. 2, a and b).

Fig. 5 shows that for -; g - and M - control

throughout the switching time for frictions units k and

1k , accordingly, the conditions are fulfilled:

0/ kaesk u , and 0)1( aeks . In

particular, when -control, the program )(tVv of the

car's motion in the time interval from 0t to 0t changes

(see Fig. 1 - the curve )(tV ). Having in mind the strategy

of inviolability of the chosen program of movement, it is

necessary to take measures to "return" to the curve )(tV .

To do this, after completing the friction trailing from the

moment 0t (

0

1

t

kR ), the engine operation mode must

instantly go to the line )( eee NN and belong to it for a

moment St , when the effects of such control will be

eliminated (point

1kR ).

Unlike the -control, during N-control there is a

slight deviation from the program )(tV (curve )(tVN in

Fig. 1). This is explained by the fact that at the beginning

of the switching the friction teailing speed k is initially

positive (to a moment 't ), but only subsequently negative

(see Fig. 5, c). Therefore, the start of the offset from the

program )(tV will occur later, and the duration of stay on

line modes )( eee NN is completed significantly earlier

(point N

kR 1 ). That is, for points N

kR 1 and

1kR ,

respectively, we have N

St <St . Consequently, the use of

dynamic controls does not perceptibly distort the motion

program, unlike others.

- Control g- Control

66 P. HASHCHUK, R.PELO

N- Control M- Control

Fig. 5. Fragments of programs for changing the modes of operation of the engine and frictions

Efficiency of gear switching laws. To substantiate the

expediency of choosing one or another method of engine

control in the process of gear shifting in the case of set

moments of its start it = 0t and the duration of work of

frictions j = , becomes possible if guided by fuel

consumption ),( jiAB tQ during the process of

reproduction of the given fragment of the program

)(tVv of motion of the vehicle. To determine fuel

consumption, we use the following formulas:

10

0

0

))()),((())(),((,0

t

t

eeet

t

t

ekektAB dtttMQdtttMQtQ

A

dtttMQdtttMQdttMQ keke

t

t

tkeke

t

t

t

t

t

t

kkkt

B

S

S

))(),(())(),(()const),(( )1()1()1()1(1)1(

0

0

10

0

, (6)

10

0

0

))()),((())(),((),( 0

t

t

eeet

t

t

ekekt

g

AB dtttMQdtttMQtQ

A

dtttMQdtttMQdtttMQ keke

t

t

tkeke

t

t

t

t

t

e

g

kekt

B

S

S

))(),(())(),(())(),(( )1()1()1()1()1(

0

0

10

, (7)

10

0

0

))()),((())(),((,0

t

t

eeet

t

t

ekekt

N

AB dtttMQdtttMQtQ

A

dtttMQdtttMQdtttMQ keke

t

t

tkeke

t

t

t

t

t

eeet

B

S

S

))(),(())(),(())()),((( )1()1()1()1(

0

0

10

, (8)

O

Me Ma M

O

e a

MekMaMe

kM

Me

e

a

t t

Me

kMMakM ,

e( )Me

kM

Ma

e( )Me

e a

econst

t

t

kMk

a

O

Me Ma M

e a

t

t

Me

MakM ,kM

( )Me

e

kMMa

( )Me

e

( )Me

e

e a

Me

tt

a

e

kM

MekMa Me

kMk

a

t

t

O

e Ma

O

e a

M

t Stt

e a

a

( )Me

e

( )Me

e

e

( )Me

e

Me

Ma

kMkMkM

MekMaMe

MakM ,Me k 1( )

kMk

kM

a

O

e a

e a

t t

t

t

Me

MeconstMaMek

kM

e

a

e a

e( =const)Me

Ma

kM

e( )Me

kM MakM ,

kMk

a

OPTIMAL LAWS OF GEAR SHIFT IN AUTOMOTIVE… 67

0

))(),((,0

t

t

ekekt

M

AB

A

dtttMQtQ

dtttMQdtttMQdtttMQ keke

t

t

tkeke

t

t

t

t

t

ekekt

B

S

S

))(),(())(),(())(,const)(( )1()1()1()1()1(

0

0

0

. (11)

The formula (6) is valid in the case of ω-control of the

thermal engine (Fig. 4a), in the case of g-control, the fuel

consumption of the engine is calculated by the formula (7)

similar to (6), but with the other limits of integration (the

value of the quantity 1 in Fig. 4, a and b are generally

not the same), calculation of fuel consumption in the case

of N - control is carried out by the formula (8). When the

fuel supply to the cylinders is stopped at the brake

operating modes of the engine, in the expressions (6) and

(7) the second conjunctions should be zero:

0))()),(((10

0

t

t

eeet dtttMQ ; (9)

and in the expression (8) the third conjunction is zero:

0))()),(((0

10

t

t

eeet dtttMQ . (10)

The relations (9) and (10) are practically always valid

when it comes to a diesel engine. In the case of M-control,

fuel consumption is determined by the formula (11), which

consists of four conjunctions (Fig. 4, d).

Universal fuel characteristic of a car. Choosing one

of the ways to control the engine - -, g-, N- or M-control

- and using equation (6), (7), (8) or (11) for any value it of the moment t of the start and any value j of the

duration of the friction trailing we can calculate the

corresponding values of the absolute fuel consumption in

the case of the implementation of the fragment AB of the

car's movement program )(tVv . On the basis of these

calculations in the coordinate system ABtOQ , it is possible

to construct a so-called fuel characteristic of a compatible

engine operation, gear box (transmission) and frictions

(fig. 6).

The fuel characteristic for a given pair of gears is

represented by the pairs of families of curves

)const,( ji

N

AB tQ and )const,( ji

g

AB tQ - in Fig. 6, a;

)const,( ji

g

AB tQ and )const,( jiAB tQ - in Fig. 6, b;

and )const,( jiAB tQ and )const,( ji

M

AB tQ - in Fig. 6,

c (upper indices N , g , , M respectively mark the

programs N -, g -, ,- M -control of the engine;

1 ii ). These series of curves belong to completely

different types of surfaces, which identify fuel

consumption in the process of different ways of engine

control (by parameters jit , , ABQ there are specific

numbers, which are not reflected in the figures for their

simplification).

Regarding the selected modes (programs, algorithms)

for controlling the thermal engine at some given 0t for

many values of the duration of the friction trailing (for

example, for8 , see Fig. 6) inequality is realized

MgN QQQQ8888

. In particular, it's easy to figure out

that in the case 8

47,28

8 N

g

Q

Q

,

63,28

8 NQ

Q

,

72,28

8 N

M

Q

Q

there is a convincing advantage of N-control

(dynamic modes). It is evident that the optimal controls,

due to the specific fuel consumption (optimal in terms of

the efficiency of the engine, which is the same) modes – g-

control for fuel economy (!) are significantly inferior to the

so-called dynamical (N-control of engine); but in

comparison with other ( - or with M-control) g-control

has not so much significant advantage, as it might seem.

Consequently, the forced methods of driving the

engine have an indisputable advantage over unforced ones.

But, analyzing fuel consumption characteristics, it is easy

to see that the surfacesM

ABQ and ABQ are interconnected

(see Fig. 6, c). A line of interconnection also have the

surfaces ABQ

g

ABQ (see Fig. 6, b). This indicates that there

are such combinations of allowable values of parameters

it and j for which the fuel consumption is either greater

or lesser, and the same, such that the grounds for refusing

to substantiate g – or -, or M way of control.

Consequently, an unconditional general conclusion about

the appropriateness of the application g, or only M-, or only

-control of the heat engine in the process of switching

gears can not be done. It turns out that with increasing the

value of the j parameter (the duration of the friction

trailing), the range of the selection of the allowable

switching moment it for all, without exception methods of

driving the engine, first increases, and then falls.

68 P. HASHCHUK, R.PELO

а б в

Fig. 6. Fuel characteristics of a car

In general, it can be noted that against the backdrop of

other types of control, the choice of the moment of start

and the duration of switching in the process of N-control

does not have a significant effect on fuel consumption. This

conclusion should be taken as positive. But still, looking at

the Fig. 6, a, we can observe that undoubtedly there are

optimal values of the quantities 00 tt

and - the

coordinates of the minimum point on the surface

),( tQQ N

ABAB . Therefore, the trailing of frictions does

not necessarily have to be as small as possible.

Adhering to the principle of (global, but not always

local) inviolability of the driver-selected driver's program,

there is no reason to distinguish between dynamic and

energy-saving transmission laws. Indeed, whatever the

driver chooses, an optimal transmission control system is

required to behave in such a way as to minimize fuel

consumption, without in any way having to worry about

how the driver works (without disturbing him). And

therefore, nothing prevents the driver from choosing an

extremely dynamic vehicle driving program if necessary,

and in this case, the transmission control system will not

have the opportunity to save fuel. From these

considerations it follows that the problem of laying a step-

by-step control system of various control algorithms (in

particular, dynamic and energy-saving) is completely

fictitious. It is easy to understand that the criteria for energy

saving and dynamism do not contradict each other.

CONCLUSIONS

Summing up the implementation of the "principal"

tasks declared here, it can be argued that the proposed

scientific hypothesis in general was true:

1. There is a certain optimum friction shaft trailing (or

any other gearshifting device), and therefore there is no

reason to advertise the seams of the switching system as a

sign of the excellence of the transmission. Because of this,

the prospects of using, for example, cam clutches can be

considered very shaky (especially out of sport). Switching

gears with cam clutches without the use of a coupler (as on

a motorcycle) for some 0.15 ... 0.20 s is unlikely to be

advantageous even when it comes to dynamism of

acceleration - there are more weighty factors (and, in

general, this issue requires a more detailed study). But the

short-live of these couplings exacerbates, among other

things, even the problem of environmental friendliness.

2. The optimal gear shift laws are identified on the

basis of not so much information that, therefore, one could

beware of the excessive complexity of the microprocessor

control system of the transmission.

3. There is no reason to oppose the energy-saving and

dynamic (sports) transmission laws. Consequently, the

motivation of poly-algorithmic control in perfect

transmissions is actually supposed.

4. The "kickdown" response per driver share

"accelerator pedal on the floor" does not belong to the

special modes of control, it is rather just one of the modes

of dynamic transmission control. And since dynamism is

not an alternative to energy saving, then there is no reason

to notice the need for "kickdown" at all.

5. Of course, one should not ignore the fact that from

time to time (including now) the primitive mechanical with

manual transmission in one way or another restore their

position in the market. And the reason is not so much in the

tastes of aggressive drivers (autosportsmen, for example,

are committed to manual transmissions), but that there is

no harmony between the promised advertising properties,

reliability, price and cost of operation. On the other hand,

even if the driver liked manual control or comfort based on

the low cost of a primitive transmission, this is by no means

the basis for "adjusting" such preferences: if manual

control is far from optimal, then for "preferences" of an

individual driver indirectly pays the whole society through

the resource and environmental problems of inefficient car

driving. The cost will no longer be particularly disturbing

if the production of a robotic transmission will have an

appropriate level of massive involvement. In general,

automation should become so perfect as to completely

eliminate the driver from the control of the transmission,

turning the ergative control system with "hints" on

automatism into a truly automatic [15].

NQ

8

gQ

8

tmint g

mint N

maxt tO maxt N g

QAB

gQ8

Q8

mintmint g

maxt ttO

QAB

MQ

8

Q8

tmaxtmintmint M

tO

QAB

OPTIMAL LAWS OF GEAR SHIFT IN AUTOMOTIVE… 69

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